Relationship between s-plane and z-plane - Electrical Engineering (EE) PDF Download

 Page 1


Digital Control Module 2 Lecture 2
Module 2: Modeling Discrete Time Systems by Pulse
Transfer Function
Lecture Note 2
1 Relationship between s-plane and z-plane
In the analysis and design of continuous time control systems, the pole-zero con?guration of the
transfer function in s-plane is often referred. We know that:
• Left half of s-plane? Stable region.
• Right half of s-plane? Unstable region.
Forrelativestabilityagainthelefthalfisdividedintoregionswherethecontrollooptransfer
function poles should preferably be located.
Similarly the poles and zeros of a transfer function in z-domain govern the performance
characteristics of a digital system.
One of the properties of F
*
(s) is that it has an in?nite number of poles, located periodically
with intervals of±mw
s
with m = 0,1,2,....., in the s-plane where w
s
is the sampling frequency
in rad/sec.
If the primary strip is considered, the path, as shown in Figure 1, will be mapped into a
unit circle in the z-plane, centered at the origin. The mapping is shown in Figure 2.
Since
e
(s+jmws)T
= e
Ts
e
j2pm
= e
Ts
= z
where m is an integer, all the complementary strips will also map into the unit circle.
1.1 Mapping guidelines
1. All the points in the left half s-plane correspond to points inside the unit circle in z-plane.
2. All the points in the right half of the s-plane correspond to points outside the unit circle.
I. Kar 1
Page 2


Digital Control Module 2 Lecture 2
Module 2: Modeling Discrete Time Systems by Pulse
Transfer Function
Lecture Note 2
1 Relationship between s-plane and z-plane
In the analysis and design of continuous time control systems, the pole-zero con?guration of the
transfer function in s-plane is often referred. We know that:
• Left half of s-plane? Stable region.
• Right half of s-plane? Unstable region.
Forrelativestabilityagainthelefthalfisdividedintoregionswherethecontrollooptransfer
function poles should preferably be located.
Similarly the poles and zeros of a transfer function in z-domain govern the performance
characteristics of a digital system.
One of the properties of F
*
(s) is that it has an in?nite number of poles, located periodically
with intervals of±mw
s
with m = 0,1,2,....., in the s-plane where w
s
is the sampling frequency
in rad/sec.
If the primary strip is considered, the path, as shown in Figure 1, will be mapped into a
unit circle in the z-plane, centered at the origin. The mapping is shown in Figure 2.
Since
e
(s+jmws)T
= e
Ts
e
j2pm
= e
Ts
= z
where m is an integer, all the complementary strips will also map into the unit circle.
1.1 Mapping guidelines
1. All the points in the left half s-plane correspond to points inside the unit circle in z-plane.
2. All the points in the right half of the s-plane correspond to points outside the unit circle.
I. Kar 1
Digital Control Module 2 Lecture 2
-ws/2
s
jw
Complementary strips Complementary strips
Primary strip
ws/2
-8
Figure 1: Primary and complementary strips in s-plane
Re z
j Im z
z-plane
1
-1
1
Figure 2: Mapping of primary strip in z-plane
3. Points on the jw axis in the s-plane correspond to points on the unit circle|z| = 1 in the
I. Kar 2
Page 3


Digital Control Module 2 Lecture 2
Module 2: Modeling Discrete Time Systems by Pulse
Transfer Function
Lecture Note 2
1 Relationship between s-plane and z-plane
In the analysis and design of continuous time control systems, the pole-zero con?guration of the
transfer function in s-plane is often referred. We know that:
• Left half of s-plane? Stable region.
• Right half of s-plane? Unstable region.
Forrelativestabilityagainthelefthalfisdividedintoregionswherethecontrollooptransfer
function poles should preferably be located.
Similarly the poles and zeros of a transfer function in z-domain govern the performance
characteristics of a digital system.
One of the properties of F
*
(s) is that it has an in?nite number of poles, located periodically
with intervals of±mw
s
with m = 0,1,2,....., in the s-plane where w
s
is the sampling frequency
in rad/sec.
If the primary strip is considered, the path, as shown in Figure 1, will be mapped into a
unit circle in the z-plane, centered at the origin. The mapping is shown in Figure 2.
Since
e
(s+jmws)T
= e
Ts
e
j2pm
= e
Ts
= z
where m is an integer, all the complementary strips will also map into the unit circle.
1.1 Mapping guidelines
1. All the points in the left half s-plane correspond to points inside the unit circle in z-plane.
2. All the points in the right half of the s-plane correspond to points outside the unit circle.
I. Kar 1
Digital Control Module 2 Lecture 2
-ws/2
s
jw
Complementary strips Complementary strips
Primary strip
ws/2
-8
Figure 1: Primary and complementary strips in s-plane
Re z
j Im z
z-plane
1
-1
1
Figure 2: Mapping of primary strip in z-plane
3. Points on the jw axis in the s-plane correspond to points on the unit circle|z| = 1 in the
I. Kar 2
Digital Control Module 2 Lecture 2
z-plane.
s = jw
z = e
Ts
= e
jwT
? magnitude = 1
1.2 Constant damping loci, constant frequency loci and constant
damping ratio loci
Constant damping loci: The real part s of a pole, s = s + jw, of a transfer function in
s-domain, determines the damping factor which represents the rate of rise or decay of time
response of the system.
• Large s represents small time constant and thus a faster decay or rise and vice versa.
• The loci in the left half s-plane (vertical line parallel to jw axis as in Figure 3(a)) denote
positive damping since the system is stable
• The loci in the right half s-plane denote negative damping.
• Constant damping loci in the z-plane are concentric circles with the center at z = 0, as
shown in Figure 3(b).
• Negative damping loci map to circles with radii > 1 and positive damping loci map to
circles with radii < 1.
s
jw
0 s2
-s1
Constant
damping loci
(a)
unit circle
s-plane
Re z
j Im z
z-plane
(b)
e
-s
1
T
e
s
2
T
Figure 3: Constant damping loci in (a) s-plane and (b) z-plane
I. Kar 3
Page 4


Digital Control Module 2 Lecture 2
Module 2: Modeling Discrete Time Systems by Pulse
Transfer Function
Lecture Note 2
1 Relationship between s-plane and z-plane
In the analysis and design of continuous time control systems, the pole-zero con?guration of the
transfer function in s-plane is often referred. We know that:
• Left half of s-plane? Stable region.
• Right half of s-plane? Unstable region.
Forrelativestabilityagainthelefthalfisdividedintoregionswherethecontrollooptransfer
function poles should preferably be located.
Similarly the poles and zeros of a transfer function in z-domain govern the performance
characteristics of a digital system.
One of the properties of F
*
(s) is that it has an in?nite number of poles, located periodically
with intervals of±mw
s
with m = 0,1,2,....., in the s-plane where w
s
is the sampling frequency
in rad/sec.
If the primary strip is considered, the path, as shown in Figure 1, will be mapped into a
unit circle in the z-plane, centered at the origin. The mapping is shown in Figure 2.
Since
e
(s+jmws)T
= e
Ts
e
j2pm
= e
Ts
= z
where m is an integer, all the complementary strips will also map into the unit circle.
1.1 Mapping guidelines
1. All the points in the left half s-plane correspond to points inside the unit circle in z-plane.
2. All the points in the right half of the s-plane correspond to points outside the unit circle.
I. Kar 1
Digital Control Module 2 Lecture 2
-ws/2
s
jw
Complementary strips Complementary strips
Primary strip
ws/2
-8
Figure 1: Primary and complementary strips in s-plane
Re z
j Im z
z-plane
1
-1
1
Figure 2: Mapping of primary strip in z-plane
3. Points on the jw axis in the s-plane correspond to points on the unit circle|z| = 1 in the
I. Kar 2
Digital Control Module 2 Lecture 2
z-plane.
s = jw
z = e
Ts
= e
jwT
? magnitude = 1
1.2 Constant damping loci, constant frequency loci and constant
damping ratio loci
Constant damping loci: The real part s of a pole, s = s + jw, of a transfer function in
s-domain, determines the damping factor which represents the rate of rise or decay of time
response of the system.
• Large s represents small time constant and thus a faster decay or rise and vice versa.
• The loci in the left half s-plane (vertical line parallel to jw axis as in Figure 3(a)) denote
positive damping since the system is stable
• The loci in the right half s-plane denote negative damping.
• Constant damping loci in the z-plane are concentric circles with the center at z = 0, as
shown in Figure 3(b).
• Negative damping loci map to circles with radii > 1 and positive damping loci map to
circles with radii < 1.
s
jw
0 s2
-s1
Constant
damping loci
(a)
unit circle
s-plane
Re z
j Im z
z-plane
(b)
e
-s
1
T
e
s
2
T
Figure 3: Constant damping loci in (a) s-plane and (b) z-plane
I. Kar 3
Digital Control Module 2 Lecture 2
Constant frequency loci: These are horizontal lines in s-plane, parallel to the real axis as
shown in Figure 4(a).
(a)
s-plane
Re z
j Im z
z-plane
(b)
0
s
j?
1
-j?
1
j?
2
Constant
frequency loci
e
j?
1
T
e
-j?
1
T
unit circle
e
j?
2
T
?
1
T
?
1
T
?
2
T
jw
Figure 4: Constant frequency loci in (a) s-plane and (b) z-plane
Corresponding Z-transform:
z = e
Ts
= e
jwT
When w = constant, it represents a straight line from the origin at an angle of ? = wT rad,
measured from positive real axis as shown in Figure 4(b).
Constant damping ratio loci: If ? denotes the damping ratio:
s = -?w
n
±jw
n
p
1-?
2
= -
w
p
1-?
2
?±jw
= -wtanß±jw
where w
n
is the natural undamped frequency and ß = sin
-1
?. If we take Z-transform
z = e
T(-wtanß+jw)
= e
-2pwtanß/ws
6 (2pw/w
s
)
Foragiven? orß, thelocusins-planeisshowninFigure5(a). Inz-plane, thecorresponding
locus will be a logarithmic spiral as shown in Figure 5(b), except for ? = 0 or ß = 0
o
and ? = 1
or ß = 90
o
.
I. Kar 4
Page 5


Digital Control Module 2 Lecture 2
Module 2: Modeling Discrete Time Systems by Pulse
Transfer Function
Lecture Note 2
1 Relationship between s-plane and z-plane
In the analysis and design of continuous time control systems, the pole-zero con?guration of the
transfer function in s-plane is often referred. We know that:
• Left half of s-plane? Stable region.
• Right half of s-plane? Unstable region.
Forrelativestabilityagainthelefthalfisdividedintoregionswherethecontrollooptransfer
function poles should preferably be located.
Similarly the poles and zeros of a transfer function in z-domain govern the performance
characteristics of a digital system.
One of the properties of F
*
(s) is that it has an in?nite number of poles, located periodically
with intervals of±mw
s
with m = 0,1,2,....., in the s-plane where w
s
is the sampling frequency
in rad/sec.
If the primary strip is considered, the path, as shown in Figure 1, will be mapped into a
unit circle in the z-plane, centered at the origin. The mapping is shown in Figure 2.
Since
e
(s+jmws)T
= e
Ts
e
j2pm
= e
Ts
= z
where m is an integer, all the complementary strips will also map into the unit circle.
1.1 Mapping guidelines
1. All the points in the left half s-plane correspond to points inside the unit circle in z-plane.
2. All the points in the right half of the s-plane correspond to points outside the unit circle.
I. Kar 1
Digital Control Module 2 Lecture 2
-ws/2
s
jw
Complementary strips Complementary strips
Primary strip
ws/2
-8
Figure 1: Primary and complementary strips in s-plane
Re z
j Im z
z-plane
1
-1
1
Figure 2: Mapping of primary strip in z-plane
3. Points on the jw axis in the s-plane correspond to points on the unit circle|z| = 1 in the
I. Kar 2
Digital Control Module 2 Lecture 2
z-plane.
s = jw
z = e
Ts
= e
jwT
? magnitude = 1
1.2 Constant damping loci, constant frequency loci and constant
damping ratio loci
Constant damping loci: The real part s of a pole, s = s + jw, of a transfer function in
s-domain, determines the damping factor which represents the rate of rise or decay of time
response of the system.
• Large s represents small time constant and thus a faster decay or rise and vice versa.
• The loci in the left half s-plane (vertical line parallel to jw axis as in Figure 3(a)) denote
positive damping since the system is stable
• The loci in the right half s-plane denote negative damping.
• Constant damping loci in the z-plane are concentric circles with the center at z = 0, as
shown in Figure 3(b).
• Negative damping loci map to circles with radii > 1 and positive damping loci map to
circles with radii < 1.
s
jw
0 s2
-s1
Constant
damping loci
(a)
unit circle
s-plane
Re z
j Im z
z-plane
(b)
e
-s
1
T
e
s
2
T
Figure 3: Constant damping loci in (a) s-plane and (b) z-plane
I. Kar 3
Digital Control Module 2 Lecture 2
Constant frequency loci: These are horizontal lines in s-plane, parallel to the real axis as
shown in Figure 4(a).
(a)
s-plane
Re z
j Im z
z-plane
(b)
0
s
j?
1
-j?
1
j?
2
Constant
frequency loci
e
j?
1
T
e
-j?
1
T
unit circle
e
j?
2
T
?
1
T
?
1
T
?
2
T
jw
Figure 4: Constant frequency loci in (a) s-plane and (b) z-plane
Corresponding Z-transform:
z = e
Ts
= e
jwT
When w = constant, it represents a straight line from the origin at an angle of ? = wT rad,
measured from positive real axis as shown in Figure 4(b).
Constant damping ratio loci: If ? denotes the damping ratio:
s = -?w
n
±jw
n
p
1-?
2
= -
w
p
1-?
2
?±jw
= -wtanß±jw
where w
n
is the natural undamped frequency and ß = sin
-1
?. If we take Z-transform
z = e
T(-wtanß+jw)
= e
-2pwtanß/ws
6 (2pw/w
s
)
Foragiven? orß, thelocusins-planeisshowninFigure5(a). Inz-plane, thecorresponding
locus will be a logarithmic spiral as shown in Figure 5(b), except for ? = 0 or ß = 0
o
and ? = 1
or ß = 90
o
.
I. Kar 4
Digital Control Module 2 Lecture 2
(a)
s-plane
Re z
j Im z
z-plane
(b)
0
s
unit circle
ß = 40
o
jw
jws
2jws
3jws
w = 0
ws
ws/2
Figure 5: Constant damping ratio locus in (a) s-plane and (b) z-plane
I. Kar 5